where \(\mathbf {C}^{\text{\textit {AB}}}\)  and \(\mathbf {D}^{\text{\textit {AB}}}\)  are the pairwise \(\ell ^{2}\) -norm matrices between the two sets of node coordinates \(\lbrace \mathbf {c}^{A}_i,\mathbf {c}^{B}_j\rbrace \) , and geodesic degrees \(\lbrace d^{A}_i,d^{B}_j\rbrace \) , as well as \(\mathbf {P}^{\text{\textit {AB}}}\) , \(\mathbf {L}^{\text{\textit {AB}}}\)  and \(\mathbf {U}^{\text{\textit {AB}}}\)  are the pairwise average symmetric distance matrices of the connecting minimal paths \(\lbrace \mathbf {p}^{A}_v,\mathbf {p}^{B}_w\rbrace \) , and the pairwise \(\ell ^{2}\) -norm matrices between the sets of the euclidean lengths \(\lbrace l^{A}_v,l^{B}_w\rbrace \)  and geodesic integral energies \(\lbrace u^{A}_v,u^{B}_w\rbrace \) , respectively.
The normalisation factors \(\sigma _{\mathbf {C},\mathbf {D},\mathbf {P},\mathbf {L},\mathbf {U}}\)  are the standard deviations estimated from the off-diagonal elements of the associated distance matrices over the considered population of graphs.
Lastly, \(\alpha _{1}\) , \(\beta _{1}\)  and \(\beta _{2}\)  weight the geometrical similarities among nodes and edges, whereas \(\alpha _{2}\)  and \(\beta _{3}\)  represent the respective geodesic trade-off.
We refer to [1]} for the composition of \(\mathbf {K}\)  from the factorised components \(\mathbf {K}_{\mathbf {n}^{\text{\textit {AB}}}}\)  and \(\mathbf {K}_{\mathbf {e}^{\text{\textit {AB}}}}\) , and for the QAP solver implementation.
